Bring to a boil; reduce heat to low, and simmer sauce while you prepare other ingredients.

Cook and stir chicken in a skillet with vegetable oil until no longer pink in the center and juices run clear, 5 to 10 minutes. Remove chicken with a slotted spoon, and set aside.

Stir shelled edamame, green onions, eggplant, 4 cloves minced garlic, mushrooms, water chestnuts, bean sprouts, cilantro, carrots, and cooked chicken into the same skillet, and cook and stir until eggplant and edamame are tender and the sauce has thickened slightly, about 15 minutes.